Question

The length of perpendicular from the origin onto the line $\vec{r} = (4 \hat{i} + 2 \hat{j} + 4 \hat{k}) + \lambda (3 \hat{i} + 4 \hat{j} - 5 \hat{k} )$ is

• A. $2 \sqrt{5}$
• B. 2
• C. $5 \sqrt{2}$
• D. 6

Answer 1

Answer: (D)

6

Answer 2

Required distance $= \frac{\left|\left(\left(4-0\right)\hat{i} + \left(2-0\right) \hat{j} + \left(4-0\right)\hat{k}\right) \times\left(3 \hat{i} + 4 \hat{j} - 5 \hat{k}\right)\right|}{ \sqrt{3^{2}+4^{2}+5^{2}}}$ $= \frac{\left|\left(4\hat{i} + 2 \hat{j} + 4 \hat{k}\right) \times\left(3\hat{i} + 4 \hat{j} - 5 \hat{k}\right)\right|}{\sqrt{50}}$ $= \frac{\left|-26 \hat{i} + 32 \hat{j} + 10\hat{k}\right|}{\sqrt{50}}$